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On the exactness of the Wu-Woodroofe approximation

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  • Klicnarová, Jana
  • Volný, Dalibor

Abstract

Let (Xi) be a stationary process adapted to a filtration , E(Xi)=0, ; by we denote the partial sums and . Wu and Woodroofe [Wei Biao Wu, M. Woodroofe, Martingale approximation for sums of stationary processes, Ann. Probab. 32 (2004) 1674-1690] have shown that if then there exists an array of row-wise stationary martingale difference sequences approximating the partial sums Sn. If then by [M. Maxwell, M. Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab. 28 (2000) 713-724] there exists a stationary martingale difference sequence approximating the partial sums Sn, and the central limit theorem holds. We will show that the process (Xi) can be found so that , constant but the central limit theorem does not hold. The linear growth of the variances is a substantial source of complexity of the construction.

Suggested Citation

  • Klicnarová, Jana & Volný, Dalibor, 2009. "On the exactness of the Wu-Woodroofe approximation," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2158-2165, July.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:7:p:2158-2165
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    References listed on IDEAS

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    1. Woodroofe, Michael, 1992. "A central limit theorem for functions of a Markov chain with applications to shifts," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 33-44, May.
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